Algebra PYQ — Page 40

The term independent of $x$ in the expression of $(1-{x}^{2}+3{x}^{3}){(\frac{5}{2}{x}^{3}-\frac{1}{5{x}^{2}})}^{11},x\neq 0$ is

The number of real solutions of the equation ${e}^{4x}+4{e}^{3x}-58{e}^{2x}+4{e}^{x}+1=0$ is _____.

The sum of the maximum and minimum values of the function $f(x)=|5x-7|+[{x}^{2}+2x]$ in the interval $[\frac{5}{4},2]$, where $[t]$ is the greatest integer $\leq t$, is ______.

The number of matrices of order $3\times 3$, whose entries are either $0$ or $1$ and the sum of all the entries is a prime number, is _______.

Let $A=[{a}_{ij}]$ be a square matrix of order $3$ such that ${a}_{ij}={2}^{j-i}$, for all $i,j=1,2,3$. Then, the matrix ${A}^{2}+{A}^{3}+\ldots +{A}^{10}$ is equal to

Let $A=[\begin{matrix}0 & -2 \\ 2 & 0\end{matrix}]$. If $M$ and $N$ are two matrices given by $M=\sum _{k=1}^{10}{A}^{2k}$ and $N=\sum _{k=1}^{10}{A}^{2k-1}$ then $M{N}^{2}$ is

The series of positive multiples of $3$ is divided into sets : ${3},{6,9,12},{15,18,21,24,27},\ldots$ Then the sum of the elements in the ${11}^{\mathrm{th}}$ set is equal to _______.

If the system of equations $x+y+z=6$ $2x+5y+\alpha z=\beta$ $x+2y+3z=14$ has infinitely many solutions, then $\alpha +\beta$ is equal to

Let the system of linear equations $x+2y+z=2$, $\alpha x+3y-z=\alpha ,-\alpha x+y+2z=-\alpha$ be inconsistent. Then $\alpha$ is equal to

The system of equations $-kx+3y-14z=25$ $-15x+4y-kz=3$ $-4x+y+3z=4$ Question: is consistent for all $k$ in the set

If the system of linear equations $2x+y-z=7$ $x-3y+2z=1$ $x+4y+\delta z=k$, where $\delta ,k\in R$ has infinitely many solutions, then $\delta +k$ is equal to

If $z=x+iy$ satisfies $|z|-2=0$ and $|z-i|-|z+5i|=0$, then

Let $S={4,6,9}$ and $T={9,10,11,\ldots ,1000}$. If $A={{a}_{1}+{a}_{2}+\ldots +{a}_{k}:k\in N,{a}_{1},{a}_{2},{a}_{3},\ldots ,{a}_{k}\in S}$ then the sum of all the elements in the set $T-A$ is equal to _______.

The domain of the function $f(x)={\mathrm{sin}}^{-1}[2{x}^{2}-3]+{\mathrm{log}}_{2}({\mathrm{log}}_{\frac{1}{2}}({x}^{2}-5x+5))$, where $[t]$ is the greatest integer function, is

Let $c,k\in R$. If $f(x)=(c+1){x}^{2}+(1-{c}^{2})x+2k$ and $f(x+y)=f(x)+f(y)-xy$, for all $x,y\in R$, then the value of $|2(f(1)+f(2)+f(3)+\ldots \ldots +f(20))|$ is equal to ______.

Let $f:R\rightarrow R$ be a function defined by $f(x)={(2(1-\frac{{x}^{25}}{2})(2+{x}^{25}))}^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, then the greatest integer less than or equal to $g(1)$ is ______.

Let $f:R\rightarrow R$ be a function defined $f(x)=\frac{2{e}^{2x}}{{e}^{2x}+e}$. Then $f(\frac{1}{100})+f(\frac{2}{100})+f(\frac{3}{100})+\ldots +f(\frac{99}{100})$ is equal to ______.

The domain of $f(x)=\frac{{\mathrm{cos}}^{-1}(\frac{{x}^{2}-5x+6}{{x}^{2}-9})}{\mathrm{log}({x}^{2}-3x+2)}$ is

For $\alpha \in N$, consider a relation $R$ on $N$ given by $R=${$(x,y):3x+\alpha y$ is a multiple of $7$}. The relation $R$ is an equivalence relation if and only if

The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is _____ .

The number of functions $f$, from the set $A={x\in N:{x}^{2}-10x+9\leq 0}$ to the set $B={{n}^{2}:n\in N}$ such that $f(x)\leq {(x-3)}^{2}+1$, for every $x\in A$, is _______.

Let the system of linear equations $x+y+az=2$ $3x+y+z=4$ $x+2z=1$ have a unique solution $(x,y,z)$. If $(\alpha ,x),(y,\alpha )$ and $(x,-y)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

For $z\in \mathbb{C}$ if the minimum value of $(|z-3\sqrt{2}|+|z-p\sqrt{2}i|)$ is $5\sqrt{2}$, then a value of $p$ is _______.

The sum of all real values of $x$ for which $\frac{3{x}^{2}-9x+17}{{x}^{2}+3x+10}=\frac{5{x}^{2}-7x+19}{3{x}^{2}+5x+12}$ is equal to